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Homogeneous Stochastic Processes Pacific Journal of Mathematics HOMOGENEOUS STOCHASTIC PROCESSES JOHN W. WOLL Vol. 9, No. 1 May 1959 HOMOGENEOUS STOCHASTIC PROCESSES* JOHN W. WOLL, JR. Summary. The form of a stationary translation-invariant Markov process on the real line has been known for some time, and these proc- esses have been variously characterized as infinitely divisible or infinitely decomposable. The purpose of this paper is to study a natural gene- ralization of these processes on a homogeneous space (X, G). Aside from the lack of structure inherent in the very generality of the spaces (X, G), the basic obstacles to be surmounted stem from the presence of non trivial compact subgroups in G and the non commutativity of G, which precludes the use of an extended Fourier analysis of characteristic functions, a tool which played a dominant role in the classical studies. Even in the general situation there is a striking similarity between homogeneous processes and their counterparts on the real line. A homogeneous process is a process in the terminology of Feller [3] on a locally compact Hausdorff space X, whose transition probabilities P(t, x, dy) are invariant under the action of elements g e G of a tran- sitive group of homeomorphisms of X, in the sense that P(t, g[x~}> g[dyj) = P(t, x, dy). It is shown that if every compact subset of X is separable or G is commutative the family of measures t~τP(t, x, •) converges to a not necessarily bounded Borel measure Qx( ) on X-{x} as £->0, meaning that for every bounded continuous, complex valued function / on X which vanishes in a neighborhood of x and is constant at infinity t-Ψ(t,x,f)-+Qx(f). In 3 we show that the paths of a separable homogeneous process are bounded on every bounded ^-interval and have right and left limits at every t with probability one. If the action of G on X is used to translate the origin of each jump to x, it is shown for suitably regular compact sets D that the probability of a jump into D while t e [0, T] is given by 1-exp {-TQX(D)}. The maps/->P(£, .,/) = (Γ,/)( ) map the Banach space, C(X), of continuous functions generated by the con- stants and functions with compact support into itself, and by a suitable normalization can be assumed strongly continuous for t > 0. Indeed, Tt is a strongly continuous semi-group. The domain D(A) of the in- finitesimal generator A of Tt admits a smoothing operation whose precise * This paper was originally accepted by the Trans. Amer. Math. Soc; received by the editors of the Trans. Amer. Math. Soc. November 5, 1956, in revised form February 24, 1958. This paper is a thesis, submitted in partial fulfillment of the requirements for the Ph. D. degree, Princeton University, June 1956. During its preparation the author was a National Science Foundation Predoctoral Fellow. 293 294 JOHN W. WOLL, Jr. nature is described in Corollary 1 of Theorem 2.2. Roughly speaking, if g 6 D(A), ε > 0, and / e C(X) we can find an h e D(A) such that \\h — /|Ioo < ε and /= h on any preassigned compact subset of interior ({x\f(x) = g(x)}). A family of measures P(x, A) which is a probability measure on the Borel subsets of X for fixed x e X, measurable in x for fixed A, and invariant under G(i.e. P(g[x], g[A]) = P(x, A)), is called a homogeneous transition probability of norm one. Such a family generates a continu- ous endomorphism /-> P( ,/) = (P/)( ) of C(X), and a homogeneous process Tt = exp{ίr(P— 1)}. This latter process is called a compound Poisson process. In 4 we study strong convergence for compound Poisson processes (Definition 4.1) and prove among other facts that every homogeneous process is a strong limit of compound Poisson proc- esses exp {tr^Pi — 1)}, and if rt < M < + oo the limit process is neces- sarily compound Poisson. If X is given the discrete topology every homogeneous process on X is compound Poisson. In case the Qx associated with P(t,x, dy) vanishes identically, or equivalently P(t, x, dy) has con- tinuous paths, we show in 5 that P(t, x, dy) is the strong limit of compound Poisson processes whose Pι{x, dz) have support arbitrarily closed to x. In 7 we study subordination of homogeneous processes as defined by Bochner [1]. By phrasing the definition in terms of a probabilistically run clock it is shown that many processes are maximal in the partial order induced by subordination. If we follow the notation in (7.2) where exp {tS(b, A, F, z)} denotes the family of characteristic functions of a homogeneous process X(t, ω) on Euclidean w-space, we obtain the following type of result. When support (F) is compact, X(t, ω) is not subordinate to any process but itself unless support (F) is also contained in the half line R+b and A = 0. In this latter case X(t, ω) is subordi- nate to the Bernoulli process Z(t, ω) = tb. Actually somewhat stronger statements can be made but they are proven only for the real line. In our notation we have not distinguished between the application of a measure μ to a function / and the measure of a measurable set E, denoting these respectively by μ(f) and μ{E). The set X-Έ is denoted by Ec and the usual convention is adopted in letting C, R, R+,Z, and Z+ represent respectively the complex numbers, the reals, the non- negative reals, the integers, and the non-negative integers. I would like to take this opportunity to express my gratitude to Professor Bochner who has patiently encouraged this work, and whose own ideas are at the base of § 7. 1. Introduction. Let G be a Hausdorff, locally compact topological group and H one of its compact subgroups, We consider G as a group HOMOGENEOUS STOCHASTIC PROCESSES 295 of homeomorphisms of the left cosets of G modulo H, which we denote by G/H, the left coset xH being mapped by a e G into the left coset axH. Moreover, if we are primarily interested in the space G/H and the action of G on this space, there is no reason why the subgroup H should play a dominant role, for the homeomorphism x -> xb of G carries the left coset xH of H into the left coset xbb^Hb of b^Hb. As far as left multiplication by elements of G is concerned this is an operator homeomorphism and G/ϋ* is equivalent to Gjb~ιHb. For this reason we use the neutral letter X for the space GjH and denote the operation of a e G on x e X by a[x]. We call the system (X, G) a homogeneous space and note that X is naturally homeomorphic to any coset space of the form G\GZ where Gz = {a e G\a[z] = z). For the sake of exposition let N(X) be the Banach space of regular bounded complex Borel measures on X; CC(X) the linear space of con- tinuous complex valued functions with compact support; CΌo(X) the closure of CC(X) in the uniform norm C(X) the Banach space of functions generated by CΌo(X) and the constant functions with the uniform norm. We use the current notation of W. Feller and denote linear trans- formations of N(X) by postmultiplication. In this notation a linear transformation T of CΌo(X) is denoted by the same letter as its adjoint transformation on N(X), viz. μT(f) = μ{Tf). By the expression μ > 0, μ e N(X), we mean μ is a real valued non-negative measure, and by the transformation La we refer to the isometries of CC(X), CΌo(X), C(X) and N(X) generated by translation of X by a e G, viz. (Laf)(x) = 1 x f(a- [x~i)f μLa(E) = μ(a[E]). When x e X we denote by δ the measure placing a unit mass at x, so that δx(E) = 1 if x e E and 0 otherwise. x aίxl For example, we shall often use the relationship S La-i= δ . Finally, when we say that a directed sequence (μq)qeQ of measures-commonly called a net in N(X) - converges weakly to μ e N(X), in symbols μQ-+ μ, we mean for every / e CC(X), μq(f) -> μ(f) as complex numbers. DEFINITION 1.1. A homogeneous transition probability is a continuous endomorphism P: N(X)-+ N(X) satisfying: (i) μ > 0 implies μP > 0 (ii) μq-*μ implies μqP-+μP; (iii) PLa = LaP. An endomorphism, P, with properties (i) and (ii) is usually called a transition probability and (iii) makes the transition probability homogeneous. When fe C^X) it follows from (ii) that SxP(f) is continuous in x. In addition δxP(f) e CΌo(X). To show this let αjz] be any directed sequence in X tending to infinity, then by virtue of the assumed com- pactness of Gzy a^D] -> infinity for any compact set D c X and 296 JOHN W. WOLL, Jr. La-ιf(y) = f(<ii[y]) -* 0 boundedly and uniformly on every compact set. It follows immediately that = δ*LarlP(f) = δΨ(Larlf) -+ 0 , proving δ*P(f) e C^X). Now for any given μ e N(X) let μq = Σ ί^V^ be a bounded, directed sequence of purely atomic measures in N(X) ap- proaching μ weakly, μq -> μ.
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